Spherical Basis Functions Research Articles

Acoustic scattering of a pair of rigid spheroids based on the T-matrix method

In this study, the T-matrix method combined with the addition theorems of spherical basis functions is applied to semi-analytically compute the underwater far-field acoustic scattering of a pair of rigid spheroids with arbitrary incident angles. The involvement of the addition theorems renders the multiple scattering fields of each spheroid to be translated into an identical origin. The accuracy and convergence property of the proposed method are verified and validated. The interference of specular reflection wave and Franz wave can be spotted from the oscillations of the form function. Furthermore, the propagation paths of specular reflection and Franz waves are quantitatively analyzed in the time domain with conclusions that the Franz waves reach the observation point subsequent to specular reflection waves and the time interval between these two wave series is equal to the time cost of the Franz waves traveling along the sphere surfaces. Finally, the effects of separation distances, aspect ratios (the ratio of the polar radius to equatorial radius), non-dimensional frequencies, and incidence angles of the plane wave on the far-field acoustic scattering of a pair of rigid spheroids are studied by the T-matrix method.

Notes on continuity result for conformable diffusion equation on the sphere: The linear case

Abstract In this article, we are interested in the linear conformable diffusion equation on the sphere. Our main goal is to establish some results on the continuity problem with respect to fractional order. The main technique is based on several evaluations on the sphere using spherical basis functions. To overcome the difficulty, we also need to use some calculations to control the generalized integrals.

Graph signal interpolation with positive definite graph basis functions

For the interpolation of graph signals with generalized shifts of a graph basis function (GBF), we introduce the concept of positive definite functions on graphs. This concept merges kernel-based interpolation with spectral theory on graphs and can be regarded as a graph analog of radial basis function interpolation in Euclidean spaces or spherical basis functions. We provide several descriptions of positive definite functions on graphs, the most relevant one is a Bochner-type characterization in terms of positive Fourier coefficients. These descriptions allow us to design GBF's and to study GBF interpolation in more detail. We are able to characterize the native spaces of the interpolants. We provide explicit estimates for the interpolation error and obtain bounds for the numerical stability. As a final application, we show how GBF interpolation can be used to get quadrature formulas on graphs.

Adaptive area-preserving parameterization of open and closed anatomical surfaces

The parameterization of open and closed anatomical surfaces is of fundamental importance in many biomedical applications. Spherical harmonics, a set of basis functions defined on the unit sphere, are widely used for anatomical shape description. However, establishing a one-to-one correspondence between the object surface and the entire unit sphere may induce a large geometric distortion in case the shape of the surface is too different from a perfect sphere. In this work, we propose adaptive area-preserving parameterization methods for simply-connected open and closed surfaces with the target of the parameterization being a spherical cap. Our methods optimize the shape of the parameter domain along with the mapping from the object surface to the parameter domain. The object surface will be globally mapped to an optimal spherical cap region of the unit sphere in an area-preserving manner while also exhibiting low conformal distortion. We further develop a set of spherical harmonics-like basis functions defined over the adaptive spherical cap domain, which we call the adaptive harmonics. Experimental results show that the proposed parameterization methods outperform the existing methods for both open and closed anatomical surfaces in terms of area and angle distortion. Surface description of the object surfaces can be effectively achieved using a novel combination of the adaptive parameterization and the adaptive harmonics. Our work provides a novel way of mapping anatomical surfaces with improved accuracy and greater flexibility. More broadly, the idea of using an adaptive parameter domain allows easy handling of a wide range of biomedical shapes.

Reconstruction of half-space boundary impedance and sound source direct radiation based on reflection coefficient estimation

When implementing the reconstruction of the sound field directly radiated from a source located in a half-space, the half-space basis functions with boundary impedance as a parameter, need to be formulated. And the boundary impedance is usually obtained via <i>in situ</i> sound impedance measurement techniques. In the reconstruction method based on the expansion in half-space spherical wave basis functions, a hologram surface and a single reference microphone placed in the near-field are used to collect sound pressures. The sound pressure at the reference microphone is first reconstructed and the error of the reconstructed pressure relative to the measured pressure is then calculated. The sound pressure reflection coefficient corresponding to the minimum error is chosen as the measure of the reflection coefficient at the measurement point. Thus, this method is applicable to reconstructing the directly radiated sound pressures without the knowledge of boundary impedance, free from the in situ sound impedance measurement techniques necessary for conventional methods. The purpose of this work is to discuss the various parameters affecting the accuracy of reconstruction. Moreover, the boundary impedance is reconstructed based on the estimation of the reflection coefficient, so that a sound impedance measurement technique implemented via the near-field acoustical holography is proposed. By taking the spherical source for example, numerical simulations are conducted to verify the proposed method of reconstructing the boundary impedance and the directly radiated sound pressures. The influences of reference microphone coordinate, the effective flow resistivity of the boundary, and the rate of decrease of porosity with depth of the boundary on the accuracy of reconstruction are quantitatively analyzed.

Reconstruction of half-space boundary impedance and sound source direct radiation based on reflection coefficient estimation

When implementing the reconstruction of the sound field radiated directly from a source located in a half-space, the half-space basis functions need to be formulated with boundary impedance as a parameter. The boundary impedance is usually obtained via <i>in situ</i> acoustic impedance measurement techniques. In a reconstruction method based on expansion in half-space spherical wave basis functions, a hologram surface and a single reference microphone placed in the near-field are used to collect sound pressures. The sound pressure at the reference microphone is first reconstructed and the error of the reconstructed pressure relative to the measured pressure is then calculated. The sound pressure reflection coefficient corresponding to the minimum error is chosen as the estimated value of the reflection coefficient at each of the measurement points. Thus, this method is applicable to reconstructing the directly radiated sound pressures without knowledge of the boundary impedance, without the <i>in situ</i> acoustic impedance measurements necessary for conventional methods. The purpose of this work is to discuss the various parameters affecting the accuracy of reconstruction. Moreover, the boundary impedance is reconstructed based on the estimation of the reflection coefficient. In this way, an acoustic impedance measurement technique implemented via the near-field acoustical holography is proposed. Taking the source to be spherical, numerical simulations are conducted to verify the proposed method of reconstructing the boundary impedance and the directly radiated sound pressures. The influences of reference microphone coordinates, the effective flow resistivity of the boundary, and the rate of decrease of porosity with depth of the boundary on the accuracy of reconstruction are quantitatively analyzed.

Quantum Hamiltonian Eigenstates for a Free Transverse Field

It is demonstrated that within the framework of the second quantization, the quantum Hamiltonian operator for a free transverse field reveals an alternative set of states satisfying the eigenstate functional equations. The construction is based on extensions of the quadratic form of the transverse Laplace operator, which are used as a source of spherical basis functions with singularity at the origin. This basis naturally replaces the basis of plane or spherical waves, which is used to separate variables with the help of the Fourier transform or transition to spherical coordinates.

Estimation of daily hydrological mass changes using continuous GNSS measurements in mainland China

Surface displacements measured by the Global Navigation Satellite System (GNSS) integrate the elastic response of the solid Earth to regional and local hydrologic loading signals and provide an opportunity for near-real-time monitoring of terrestrial water storage variations. Here, we estimate the spectral information of hydrology-induced vertical surface deformation based on spherical Slepian basis functions and recover the daily changes in continental water storage over mainland China from January 2010 to December 2019. Our inversion results, depending on a sparsely-distributed GNSS network, indicate the largest seasonal fluctuation of hydrological mass loads in southwestern China. The GNSS-inferred equivalent water height has an annual amplitude of up to 314 mm, which is greater than 243 mm from the Gravity Recovery and Climate Experiment (GRACE) and 150 mm from the Global Land Data Assimilation System (GLDAS). We also find notable seasonal water oscillations of 153–166 mm in the middle and lower Yangtze River Basin. This feature appears in the GNSS inversion model and GRACE Mascon solutions but is invisible in the GLDAS model, mostly indicating the existence of significant changes in surface water storage that is unmodeled in the GLDAS model. There is board agreement in the water estimates derived from multiple data sets in South China, in contrast to weak temporal coherence of various water estimates in North China. We also demonstrate the ability of GNSS for tracking the drought and wet periods, which can serve as an independent means to characterize hydrological extremes. Our study emphasizes that a sparse GNSS network can still be considered as a complementary tool to map large-scale spatiotemporal variations of water storage and benefit the hydrological community for assessing water storage changes and hydrological dynamics.

Constrained spherical deconvolution of nonspherically sampled diffusion MRI data.

Constrained spherical deconvolution (CSD) of diffusion‐weighted MRI (DW‐MRI) is a popular analysis method that extracts the full white matter (WM) fiber orientation density function (fODF) in the living human brain, noninvasively. It assumes that the DW‐MRI signal on the sphere can be represented as the spherical convolution of a single‐fiber response function (RF) and the fODF, and recovers the fODF through the inverse operation. CSD approaches typically require that the DW‐MRI data is sampled shell‐wise, and estimate the RF in a purely spherical manner using spherical basis functions, such as spherical harmonics (SH), disregarding any radial dependencies. This precludes analysis of data acquired with nonspherical sampling schemes, for example, Cartesian sampling. Additionally, nonspherical sampling can also arise due to technical issues, for example, gradient nonlinearities, resulting in a spatially dependent bias of the apparent tissue densities and connectivity information. Here, we adopt a compact model for the RFs that also describes their radial dependency. We demonstrate that the proposed model can accurately predict the tissue response for a wide range of b‐values. On shell‐wise data, our approach provides fODFs and tissue densities indistinguishable from those estimated using SH. On Cartesian data, fODF estimates and apparent tissue densities are on par with those obtained from shell‐wise data, significantly broadening the range of data sets that can be analyzed using CSD. In addition, gradient nonlinearities can be accounted for using the proposed model, resulting in much more accurate apparent tissue densities and connectivity metrics.

Generation of energy-minimizing point sets on spheres and their application in mesh-free interpolation and differentiation

It is known that discrete sets of uniformly distributed points on the hypersphere $\mathbb {S}^{d}\subset \mathbb {R}^{d+1}$ can be obtained from minimizing the energy functional corresponding to Riesz s-kernels $k_s(\boldsymbol {x},\boldsymbol {y})=\lVert \boldsymbol {x}-\boldsymbol {y}\rVert ^{-s}$ (s > 0) or the logarithmic kernel $k_{\log }(\boldsymbol {x},\boldsymbol {y})=-\log \lVert \boldsymbol {x}-\boldsymbol {y}\rVert +\log 2$. We prove the same for the kernel $k_{\log }(\boldsymbol {x},\boldsymbol {y})=\lVert \boldsymbol {x}-\boldsymbol {y}\rVert (\log {\frac {\lVert \boldsymbol {x}-\boldsymbol {y}\rVert }{2}}-1)+2$ which is a front-extension of the sequence of derivatives $k_{\log }, k_{1}, k_{2}, k_{3}, \dots $, up to sign and constants. The boundedness of the kernel simplifies the classical potential-theoretical proof of the asymptotic uniformity of the point distributions. Still, the property of a singular derivative for x → y is preserved, with the physical interpretation of infinite repulsive forces for touching particles. The quality of the resulting point distributions is exemplary compared with that of Riesz- and classical logarithmic point sets, and found to be competitive. Originally motivated by problems of high-dimensional data, the applicability of $\log $-optimal point sets with a novel concentric interpolation and differentiation scheme is demonstrated. The method is significantly optimized by the introduction of symmetrized kernels for both the generation of the minimum energy points and the spherical basis functions. Both the point generation and the Concentric Interpolation software are available as Open Source software and selected point sets are provided.

A high-order meshless Galerkin method for semilinear parabolic equations on spheres

We describe a novel meshless Galerkin method for numerically solving semilinear parabolic equations on spheres. The new approximation method is based upon a discretization in space using spherical basis functions in a Galerkin approximation. As our spatial approximation spaces are built with spherical basis functions, they can be of arbitrary order and do not require the construction of an underlying mesh. We will establish convergence of the meshless method by adapting, to the sphere, a convergence result due to Thomee and Wahlbin. To do this requires proving new approximation results, including a novel inverse or Nikolskii inequality for spherical basis functions. We also discuss how the integrals in the Galerkin method can accurately and more efficiently be computed using a recently developed quadrature rule. These new quadrature formulas also apply to Galerkin approximations of elliptic partial differential equations on the sphere. Finally, we provide several numerical examples.

Utilizing new spherical Hankel shape functions to reformulate the deflection, free vibration, and buckling analysis of Mindlin plates based on finite element method

In this study, a new class of shape functions, namely spherical Hankel shape functions, are derived and applied to reformulate the deflection, free vibration, and buckling of Mindlin plates based on finite element method (FEM). In this way, adding polynomial terms to the functional expansion, in which just spherical Hankel radial basis functions (RBFs) are used, leads to obtaining spherical Hankel shape functions. Accordingly, the employment of polynomial and spherical Bessel function fields together results in achieving more robustness and effectiveness. Spherical Hankel shape functions benefit from some useful properties, including infinite piecewise continuity, partition of unity, and Kronecker delta property. In the end, the accuracy of the proposed formulation is investigated through several numerical examples for which the same degrees of freedom are selected in both the presented formulation and the classical finite element method. Finally, it can be concluded that a higher accuracy is reachable by utilizing spherical Hankel shape functions in comparison with the Lagrangian FEM.

Numerical calculation of the Spherical Bessel Transform from Gaussian quadrature in the complex-plane

A numerical method for the calculation of spherical Bessel transforms via Gaussian quadrature in the complex plane is presented. The method is evaluated by transforming Slater and Gaussian-type spherical coordinate basis functions, used in electronic structure calculations, from position space to momentum space. The feasibility and efficiency of the method is explored for different regions of momenta and different orders of the spherical Bessel transform. The results illustrate that in general the method performs very well in the large p regions when applied to the Slater-type functions. On the other hand, a parameter has to be introduced for the application to Gaussian-type functions in order to avoid cancellation. In this case, the method performs well but accuracy is lost at larger p. Use of the parameter in application to Slater functions yields accurate results for all regions of p.

The improvement of numerical modeling in the solution of incompressible viscous flow problems using finite element method based on spherical Hankel shape functions

SummaryIn this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.

GPS Recovery of Daily Hydrologic and Atmospheric Mass Variation: A Methodology and Results From the Australian Continent

AbstractWe present a methodology to invert a regional set of vertical displacement data from Global Positioning System (GPS) to determine the surface mass redistribution. It is assumed that GPS deformation is a result of the Earth's elastic response to the surface mass load of hydrology, atmosphere, and/or ocean. We develop an algorithm to estimate the spectral information of displacements from “regional” GPS data through regional spherical (Slepian) basis functions and apply the load Love numbers to estimate the mass load. The same approach is applied to determine global mass changes from “global” geopotential change data of Gravity Recovery and Climate Experiment (GRACE). We rigorously examine all systematic errors caused by various truncations (spherical harmonic series and Slepian series) and the smoothing constraint applied to the GPS inversion. We demonstrate the technique by processing 16 years of daily vertical motions determined from 114 GPS stations in Australia. The GPS‐inverted surface mass changes are validated against GRACE data, atmosphere and ocean models, and a land surface model. Seasonal and interannual terrestrial mass variations from GPS are in good agreement with GRACE data and the water storage models. The GPS recovery compares better with the water storage model around the smaller coastal basins than two different GRACE solutions. The submonthly mass changes from GPS provide meaningful results agreeing with atmospheric mass changes in central Australia. Finally, it is suggested to integrate GPS and GRACE data to draw a comprehensive picture of daily mass changes on different continents.

Protein–protein docking by fast generalized Fourier transforms on 5D rotational manifolds

Energy evaluation using fast Fourier transforms (FFTs) enables sampling billions of putative complex structures and hence revolutionized rigid protein-protein docking. However, in current methods, efficient acceleration is achieved only in either the translational or the rotational subspace. Developing an efficient and accurate docking method that expands FFT-based sampling to five rotational coordinates is an extensively studied but still unsolved problem. The algorithm presented here retains the accuracy of earlier methods but yields at least 10-fold speedup. The improvement is due to two innovations. First, the search space is treated as the product manifold [Formula: see text], where [Formula: see text] is the rotation group representing the space of the rotating ligand, and [Formula: see text] is the space spanned by the two Euler angles that define the orientation of the vector from the center of the fixed receptor toward the center of the ligand. This representation enables the use of efficient FFT methods developed for [Formula: see text] Second, we select the centers of highly populated clusters of docked structures, rather than the lowest energy conformations, as predictions of the complex, and hence there is no need for very high accuracy in energy evaluation. Therefore, it is sufficient to use a limited number of spherical basis functions in the Fourier space, which increases the efficiency of sampling while retaining the accuracy of docking results. A major advantage of the method is that, in contrast to classical approaches, increasing the number of correlation function terms is computationally inexpensive, which enables using complex energy functions for scoring.

Combination of various observation techniques for regional modeling of the gravity field

AbstractModeling a very broad spectrum of the Earth's gravity field needs observations from various measurement techniques with different spectral sensitivities. Typically, high‐resolution regional gravity data are combined with low‐resolution global observations. To exploit the gravitational information as optimally as possible, we set up a regional modeling approach using radial spherical basis functions, emphasizing the strengths of various data sets by the flexible combination of high‐ and middle‐resolution terrestrial, airborne, shipborne, and altimetry measurements. The basis functions are defined and located in the region of interest in such a manner, which the highest measure of information of the input data is captured. Any functional of the Earth's gravity field can be derived, as, e.g., quasi‐geoid heights or gravity anomalies. Here we present results of a study area in Northern Germany. A comprehensive cross validation to external observation data delivers standard deviations less than 5 cm. Differences to an existing regional quasi‐geoid model count on average ±6 cm and proof the plausibility of our solution. The comparison with existing global models reaches higher standard deviations for the more sensitive gravity anomalies as for quasi‐geoid heights, showing the additional value of our solution in the high frequency domain. Covering a broad frequency spectrum, our regional models can be used as basis for various applications, such as refinement of global models, national geoid determination, and detection of mass anomalies in the Earth's interior.

A radiance cache method for highly glossy surfaces

Radiance caching methods have proven to be efficient for global illumination. Their goal is to compute precisely illumination values (incident radiance or irradiance) at a reasonable number of points lying on the scene surfaces. These points, called records, are stored in a cache used for estimating illumination at other points in the scene. Unfortunately, with records lying on glossy surfaces, the irradiance value alone is not sufficient to evaluate the reflected radiance; each record should also store the incident radiance for all incident directions. Memory storage can be reduced with projection techniques using spherical harmonics or other basis functions. These techniques provide good results for low shininess BRDFs. However, they get impractical for shininess of even moderate value, since the number of projection coefficients increases drastically. In this paper, we propose a new radiance caching method that handles highly glossy surfaces while requiring a low memory storage. Each cache record stores a coarse representation of the incident illumination thanks to a new data structure, called Equivalent Area light Sources, capable of handling fuzzy mirror surfaces. In addition, our method proposes a new simplification of the interpolation process, since it avoids the need for expressing and evaluating complex gradients.

A more stable transition matrix for acoustic target scattering by elongated objects.

The transition (T) matrix of Waterman has been very useful for computing fast, accurate acoustic scattering predictions for axisymmetric elastic objects, but this technique is usually limited to fairly smooth objects that are not too aspherical unless complex basis functions or stabilization schemes are used. To ease this limitation, a spherical-basis formulation adapted from approaches proposed recently by Waterman [J. Acoust. Soc. Am. 125(1), 42-51 (2009)] and Doicu, Eremin, and Wriedt [Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources (Academic, London, 2000)] is suggested. This is implemented by simply transforming the high-order outgoing spherical basis functions within standard T-matrix formulations to low-order functions distributed along the object's symmetry axis. A free-field T matrix is produced in a nonstandard form, but computations with it become much more stable for elongated aspherical elastic shapes. Some advantages of this approach over the approaches of Waterman and Doicu, Eremin, and Wriedt are noted, and sample calculations for a 10:1 Al prolate spheroid and a 10:1 Al superspheroid of order 10 are given to demonstrate the enhanced stability.

A low frequency elastodynamic fast multipole boundary element method in three dimensions

This paper presents a fast multipole boundary element method (FMBEM) for the 3-D elastodynamic boundary integral equation in the `low frequency' regime. New compact recursion relations for the second-order Cartesian partial derivatives of the spherical basis functions are derived for the expansion of the elastodynamic fundamental solutions. Numerical solution is achieved via a novel combination of a nested outer---inner generalized minimum residual (GMRES) solver and a sparse approximate inverse preconditioner. Additionally translation stencils are newly applied to the elastodynamic FMBEM and an implementation of the 8, 4 and 2-box stencils is presented, which is shown to reduce the number of translations per octree level by up to $$60\,\%$$60%. This combination of strategies converges 2---2.5 times faster than the standard GMRES solution of the FMBEM. Numerical examples demonstrate the algorithmic and memory complexities of the model, which are shown to be in good agreement with the theoretical predictions.